| L(s) = 1 | + 3-s − 5-s − 7-s − 2·9-s − 3·11-s − 4·13-s − 15-s − 4·19-s − 21-s + 25-s − 5·27-s + 2·31-s − 3·33-s + 35-s + 37-s − 4·39-s + 3·41-s + 2·43-s + 2·45-s + 3·47-s − 6·49-s − 9·53-s + 3·55-s − 4·57-s + 2·61-s + 2·63-s + 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.904·11-s − 1.10·13-s − 0.258·15-s − 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.962·27-s + 0.359·31-s − 0.522·33-s + 0.169·35-s + 0.164·37-s − 0.640·39-s + 0.468·41-s + 0.304·43-s + 0.298·45-s + 0.437·47-s − 6/7·49-s − 1.23·53-s + 0.404·55-s − 0.529·57-s + 0.256·61-s + 0.251·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.882130447506378962978172934692, −9.045668969928086126901178766839, −8.153762341243578669363344924181, −7.57337471682987225514682928288, −6.48137662811284452613419092940, −5.38893234254047555733787644652, −4.36492457617219525453698113063, −3.12763491076479958195283824705, −2.33839293362589242917551382192, 0,
2.33839293362589242917551382192, 3.12763491076479958195283824705, 4.36492457617219525453698113063, 5.38893234254047555733787644652, 6.48137662811284452613419092940, 7.57337471682987225514682928288, 8.153762341243578669363344924181, 9.045668969928086126901178766839, 9.882130447506378962978172934692
